Motivation

In [2] we have a derivation of the Fresnel equations for the TE and TM polarization modes. Can we do this for an arbitrary polarization angles?

Setup

The task at hand is to find evaluate the boundary value constraints. Following the interface plane conventions of [1], and his notation that is

I’ll work here with a phasor representation directly and not bother with taking real parts, or using tilde notation to mark the vectors as complex.

Our complex magnetic field phasors are related to the electric fields with

Referring to figure (see pdf) shows the geometrical task to tackle, since we’ve got to express all the various unit vectors algebraically. I’ll use Geometric Algebra here to do that for its compact expression of rotations. With

Figure: See pdf: Reflection and transmission of light at an interface

we can express each of the vector directions by inspection. Those are

Similarly, the perpendiculars are

In [1] problem 9.14 we had to show that the polarization angles for normal incident () must be the same due to the boundary constraints. Can we also tackle that problem for both this more general angle of incidence and a general polarization? Let’s try so, allowing temporarily for different polarizations of the reflected and transmitted components of the light, calling those polarization angles , , and respectively. Let’s set the polarization aligned such that , are aligned with the and directions respectively, so that the generally polarized phasors are

We are now set to at least express our boundary value constraints

Let’s try this in a couple of steps. First with polarization angles set so that one of the fields lies in the plane of the interface (with both variations), and then attempt the general case, first posing the problem in the tranditional way to see what equations fall out, and then using superposition.

Before doing so, let’s introduce a bit of notation to be used throughout. When we wish to refer to all the fields or angles, for example, then we’ll write where . Similarily, to refer to just the incident and transmitted components (or angles) we’ll use where . Following [1] we’ll also write

Question: Sanity check. Verify for parallel to the interface.

Answer

For the polarization () our phasors are

Our boundary value constraints then become

With substitution this is

Evaluating the grade selections we have a separation into an analogue of real and imaginary parts for

With and 1.2.12b becomes

so that we find 1.2.12b and 1.2.12c are dependent. We are left with a pair of equations

Adding and subtracting we have

with a final rearrangement to yield

Using the and notation above we have

Question: Sanity check. Verify for parallel to the interface.

Answer

As a second sanity check let’s rotate our field polarizations by applying a rotation () so that

This time we have and . Our boundary value equations become

This second equation 1.2.19b is a identity, and the remaining after substitution are

Simplifying we have

We expect an equality

Noting that we find that to be true

we see that 1.2.21a and 1.2.21c are dependent. We are left with the system

with solution

Question: General case. Arbitrary polarization angle.

Determine the set of simulaneous equations that would have to be solved for if the incident polarization angle was allowed to be neither TE nor TM mode.

Answer

Substituting our vector expressions into the boundary value constraints we have

With we want to expand some intermediate multivector products

Our boundary value conditions are then

Note that the wedge product equations above have been separated into and components, yielding two equations each. Because of 1.2.21c, we see that 1.2.31 and 1.2.31 are dependent. Also, as demonstrated in 1.2.12d we see that 1.2.31 and 1.2.31 are also dependent. We can therefore consider only the last four equations (and still have additional linear dependencies to be discovered.)

Let’s write these as

Observe that if (killing all the sine terms) we recover 1.2.14, and with (killing all the cosines) we recover 1.2.24.

Now, if we’ve got a different story. Specifically it appears that should we wish to solve for the reflected and transmitted magnitudes, we also have to simulaneously solve for the polarization angles in the reflected and transmitted directions. This is now a problem of solving four simulaneous equations in two linear and two non-linear variables.

Does it make sense that we would have polarization rotation should our initial polarization angle be rotated? I think so. In dicusssing this problem with Prof Thywissen, he strongly suggested treating the problem as a superposition of two light waves. If we consider that, even without attempting to solve the problem, we see that we must have different reflected and transmitted magnitudes associated with the pair of incident waves since we have to calculate each of these with different Fresnel equations. This would have an effect of scaling and rotating the superimposed reflected and transmitted waves.

Question: General case using using superposition

Using superposition determine the Fresnel equations for an arbitrary incident polarization angle. This should involve solving for both the magnitude and the polarization angle of the reflected and transmitted rays.

Answer

For a polarization of and respectively, we have from problems \ref{fresnelAlternatePolarization:pr1-Answer} and \ref{fresnelAlternatePolarization:pr2-Answer}, or from 1.2.32 we have

We can use these results to consider a polarization of as illustrated in figure (see pdf)

Figure: see pdf: Polarization of incident field to be considered

Our incident, reflected, and transmitted fields are

However, and leaving us with

We find that the reflected and transmitted polarization angles are respectively

I’ve got a scenerio where it appears that the last stack variable declared appears to be have been corrupted (the highest order 32-bits of this 64-bit integer look like they’ve been zeroed). That got me wondering how far a calling function would have to corrupt to muck up this variable. Here’s what I wrote to test this:

So, it looks like I need about at least a (0x88-0x50 =) 56 byte corruption to do the job.

A quirk: Also see how the variables in my function actually got laid out in reverse address order on the stack. I’d not have expected that. However, since I don’t really have any reason to expect any specific stack layout so perhaps I shouldn’t be surprised.

This isn’t an nm equivalent, instead is more like the AIX command to dump just the exported symbols from a shared object (dump -TvHX32_64), but enough to tell me that I shouldn’t have a link error this iteration of my build.

I found it curious that ‘dumpbin /symbols’ didn’t produce any output for this dll, as is does for a .obj file, and don’t really know what the reason for that is.

Motivation

Study of reflection and transmission of radiation in isotropic, charge and current free, linear matter utilizes the plane wave solutions to Maxwell’s equations. These have the structure of phasor equations, with some specific constraints on the components and the exponents.

These constraints are usually derived starting with the plain old vector form of Maxwell’s equations, and it is natural to wonder how this is done directly using Geometric Algebra. [1] provides one such derivation, using the covariant form of Maxwell’s equations. Here’s a slightly more pedestrian way of doing the same.

Maxwell’s equations in media

We start with Maxwell’s equations for linear matter as found in [2]

We merge these using the geometric identity

where is the 3D pseudoscalar , to find

We want dimensions of for the derivative operator on the RHS of 1.2.3b, so we divide through by for

This can now be added to 1.2.3a for

This is Maxwell’s equation in linear isotropic charge and current free matter in Geometric Algebra form.

Phasor solutions

We write the electromagnetic field as

so that for vacuum where we have the usual . Assuming a phasor solution of

where is allowed to be complex, and the actual field is obtained by taking the real part

Note carefully that we are using a scalar imaginary , as well as the multivector (pseudoscalar) , despite the fact that both have the square to scalar minus one property.

We now seek the constraints on , , and that allow this to be a solution to 1.2.5

As usual in the non-geometric algebra treatment, we observe that any such solution to Maxwell’s equation is also a wave equation solution. In GA we can do so by right multiplying an operator that has a conjugate form,

where is the speed of the wave described by this solution.

Inserting the exponential form of our assumed solution 1.3.7 we find

which implies that the wave number vector and the angular frequency are related by

Our assumed solution must also satisfy the first order system 1.3.9

The constraints on must then be given by

With

we must then have all grades of the multivector equation equal to zero

Writing out all the geometric products, noting that commutes with all of , , and and employing the identity we have

This is

This and 1.3.12 describe all the constraints on our phasor that are required for it to be a solution. Note that only one of the two cross product equations in are required because the two are not independent. This can be shown by crossing with 1.3.18b and using the identity

One can find easily that 1.3.18b and 1.3.18c provide the same relationship between the and components of . Writing out the complete expression for we have

Since , this is

Had we been clever enough this could have been deduced directly from the 1.3.14 directly, since we require a product that is killed by left multiplication with . Our complete plane wave solution to Maxwell’s equation is therefore given by